"There are a lot of smug haters out there who bag on my choice of a name, but I don't care about what they think," she wrote. "I want my daughter to be proud of who she is and learn to speak up and stand up for herself at a young age."

"I spent most of my life being a people-pleaser who worried about what other people thought or thought was cool and I don't want that for her," she continued.

The former Playboy playmate is also confident that Rainbow's name will not have a negative impact on her little girl's life.

"I have always loved the name Rainbow," she wrote. "There was a girl in my school a few years younger than me named Rainbow and I was so envious of her name because it was so pretty and unusual. She was perfectly normal, well-adjusted, sporty girl, by the way, so I'm not worried about my daughter being 'traumatized' by having an unusual name."

Madison, 33, gave birth to her tiny bundle of joy on Tuesday, March 5 (father is Madison's boyfriend, Pasquale Rotella), and said that she has "never been happier" than she is now for her "perfect daughter."